A 8 kg crate is placed at rest on an incline plane (s=0.8, k=0.5), whose angle is 60o. It then slides 3 m down the incline.
work done by gravity: ?J
work done by friction force: ?J
Using your results from part a, find the speed of the crate after sliding this distance.
The normal force on the plane from weight is mgCosTheta. The component of force due to weight down the plane is mgSinTheta.
Work down the plane is Ffriction*distance, or mg*mu*SinTheta*distance.
Work done by gravity is mg*distancedownplane*SinTheta Think that out...
To find the work done by gravity, we can use the formula:
Work = Force * Distance * cos(angle)
The force due to gravity can be calculated using the formula:
Force_gravity = mass * gravity
where mass is the mass of the crate, which is 8 kg, and gravity is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Using these values, we can calculate the force due to gravity:
Force_gravity = 8 kg * 9.8 m/s^2 = 78.4 N
Now we can calculate the work done by gravity:
Work_gravity = Force_gravity * Distance * cos(angle)
= 78.4 N * 3 m * cos(60°)
≈ 135.86 J
Therefore, the work done by gravity is approximately 135.86 J.
To find the work done by the friction force, we can use the formula:
Work = Force * Distance
The force of friction can be calculated using the formula:
Force_friction = coefficient of friction * normal force
The normal force can be calculated using the formula:
Normal force = mass * gravity * cos(angle)
Using the given coefficient of friction (k = 0.5), we can calculate the force of friction:
Force_friction = 0.5 * (8 kg * 9.8 m/s^2 * cos(60°))
= 19.6 N
Now we can calculate the work done by the friction force:
Work_friction = Force_friction * Distance
= 19.6 N * 3 m
= 58.8 J
Therefore, the work done by the friction force is 58.8 J.
To find the speed of the crate after sliding 3 m down the incline, we can use the principle of conservation of mechanical energy. The mechanical energy of the crate can be calculated using the formula:
Mechanical energy = Potential energy + Kinetic energy
Potential energy = mass * gravity * height
In this case, the height of the incline is given by:
Height = Distance * sin(angle)
= 3 m * sin(60°)
≈ 2.598 m
Therefore, the potential energy can be calculated as:
Potential energy = 8 kg * 9.8 m/s^2 * 2.598 m
≈ 202.2 J
Since the crate starts at rest and there is no additional energy input, all the potential energy will be converted to kinetic energy.
Therefore, the kinetic energy can be calculated as:
Kinetic energy = Mechanical energy - Potential energy
= 135.86 J + 58.8 J - 202.2 J
≈ -7.54 J (negative value due to the loss of energy by friction)
Since kinetic energy is equal to (1/2) * mass * velocity^2, we can rearrange the equation to find the speed (velocity) of the crate:
Velocity = sqrt((2 * Kinetic energy) / mass)
= sqrt((2 * -7.54 J) / 8 kg)
≈ 1.73 m/s
Therefore, the speed of the crate after sliding 3 m down the incline is approximately 1.73 m/s.